A ray of light from a denser medium strikes a rarer medium. The reflected and refracted rays make an angle of$90°$ with each other. The angles of reflection and refraction are r and${\text{r}}^{\text{1}}$ . The critical angle would be

$i=r$

${\text{r}}^{\text{1}}=90-\text{r}$

${\mu }_2\sin i={\mu }_1\sin {\text{r}}^{\text{1}}$                    

${\mu }_2\sin \text{r}={\mu }_1\sin \left(90-\text{r}\right)$

$\frac{{\mu }_2}{{\mu }_1}\sin \text{r}=\cos \text{r}$

$\frac{\sin r}{\cos r}=\frac{{\mu }_1}{{\mu }_2}$

$\tan r=\frac{{\mu }_1}{{\mu }_2}$

$\Rightarrow \frac{{\mu }_2}{{\mu }_1}=\frac{1}{\tan \text{r}}$

$\text{But}\frac{{\mu }_2}{{\mu }_1}=\frac{1}{\sin c}$

$\Rightarrow \sin \text{c}=\tan \text{r}$                        

$\text{c}={\text{sin}}^{-1}\left(\tan \text{r}\right)$